1 On Stochastic Processes Driven By Ballistic Noise ...

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On Stochastic Processes Driven By Ballistic Noise Sources

M.-O. Hongler1, R. Filliger2, Ph. Blanchard3 and J. Rodriguez3 Faculté des Sciences et Techniques de l’Ingénieur,

1

Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, i-REX Institute, Bern University of Applied Sciences, CH-2501 Biel,

2

Fakultät für Physik und BiBos, Universität Bielefeld, D-33615 Bielefeld.

3

Email:  [email protected];  [email protected]

Abstract: Among the rich collection of noise sources that can be used to drive stochastic differential equations (SDE), we here focus on Markov processes that themselves are functions of the standard Brownian motion and of the Telegrapher’s process. Our present noise sources can be viewed as lumped versions of Markov processes with enlarged states spaces. The resulting Markov processes exhibit ballistic non-Gaussian noise features with long time range correlations. When used as simple additive noise sources in SDEs, these ballistic fluctuations may generate noise induced structures (i.e., noise induced transitions), a behavior that is usually only observed for multiplicative de-correlated noise sources. Our ballistic noise sources are then used in a selection of applications taken from control theory and multi-agents systems for which exact analytic results are derived.

1.1  Introduction and Basic Motivation Noise in dynamical systems arise naturally either from the fluctuations of control parameters, (this is often referred as external noise) or as the effective action of numerous fast evolving degrees of freedom affecting slowly relaxing variables in complex systems, (referred in this case as internal noise). In both situations randomness is added to the deterministic evolution of the dynamically relevant variables. It results SDEs or Langevin equations having the form: *

partially supported by the FNSR (Fonds National Suisse pour la Recherche).

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dX t = f ( X t , t )dt + σ( X t )dZ t ,   X 0 = x0 ,



(1.1)

where the drift function f characterizes the deterministic evolution and where σ measures the noise amplitude of the centered noise source Zt. For simplicity, we here assume Eq.(1.1) to be a real valued equation (with f and σ satisfying standard Lipschitz and boundedness conditions, see e.g. [1.1] Chapt. 4.3) and therefore the solution Zt is a scalar, continuous time stochastic process. The by far most widely used noise source for modeling purposes is White Gaussian Noise (WGN). WGN is analytically tractable and adequate whenever the system dynamics is slow compared to the relaxation time of the system noise. By WGN, we mean the generalized Gaussian process which can be constructed as a probability measure on the [0,∞ ] space of tempered distributions (and not as a measure on the much smaller space  like ordinary processes). Informally, WGN is given by the time derivative of the Wiener process Wt, i.e., the velocity of the Brownian Motion (BM) and we write dZ t = dWt for the noise source. WGN is centered, 〈 dWt 〉 = 0 , and delta-correlated: 〈 dWt , dWs 〉 = δ (t − s ), (1.2) (here 〈⋅〉 denotes expectation i.e., averaging with respect to statistical ensemble). Taking the WGN in Eq.(1.1) ensures that X t is a Markov diffusion process with trajectories which are almost surely continuous. The statistic evolution of X t is fully characterized by the associated transition probability density (TPD) P( x, t | x0 , t0 ) , t ≥ t0 , x0 ∈ , which is given by the normalized solution to the Fokker-Planck equation  σ 2 ( x)  ∂t P( x, t | x0 , t0 ) = −∂ x  f ( x, t ) P( x, t | x0 , t0 )  + ∂ x, x  P( x, t | x0 , t0 )  .  2 

(1.3)

For the scalar process Eq.(1.1) and for reflecting boundary conditions, one can directly write the stationary measure Ps ( x) solving Eq.(1.3) as:  x f ( x ′ )  (1.4) exp dx ′  , ∫ 2 2 2 σ ( x)  σ ( x ′ )  where N is the normalization factor. For constant and small noise amplitude σ the stationary distribution is sharply peaked around the deterministic stationary points x* defined by f (x*) = 0. In general however, (i.e., for non scalar situations or general boundary conditions), the invariant measure solving the time independent Eq.(1.3) cannot be integrated in closed form and one relies on various approximation methods, such as perturbation theory, WKB approximation, numerics (see e.g., [1.1] sect. 5.2).

Ps ( x) =



Obviously, the absence of time correlations makes WGN a stylized representation of reality, as finite correlations necessarily exist for any finite energy noise sources. If the relaxation time of the noise is not negligible with respect to the system state, one has to use colored noise instead of white noise. The simplest and most commonly used colored noise source, is the (centered) Ornstein-Uhlenbeck (OU) process [1.3]:

On Stochastic Processes Driven by Ballistic Noise Sources

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dU t = − λU t dt + λσdWt

(1.5)



with λ and σ constants. This stationary Gauss-Markov process has exponentially decaying time-correlations of the form: λσ 2 − λ|t − s| (1.6) e . 2 With the chosen representation, one observes that for λ → ∞ , the δ -correlated WGN is recovered. Introducing Eq. (1.5) as a noise source into Eq. (1.1), one considers the extended system:



〈U t , U s 〉 =

dX t = f ( X t , t )dt + dZ t , X 0 = x0 ,  dZ t = Udt , dU = − λU dt + σdW , U = 0. t t 0  t



(1.7)

While the process ( X t , U t ) is Markovian, the process Xt alone is not. The common strategy to gain statistical insight on the dynamics Xt is to compute the X-marginal transition density of the joint diffusion process ( X t , U t ) whose dynamics is fully characterized by its transition probability density P = P( x, u , t | x0 , 0, t0 ) . The latter is the normalized solution to ∂t P =  ( P) with the diffusion kernel  :

 (⋅) = −∂ x [ f ( x, t ) + u ] (⋅) + ∂u λu (⋅) + σ 2 2∂u ,u (⋅).

(1.8)

Except for quasi-linear dynamics (i.e., of the form f ( X t , t ) = a (t ) X t + b(t ) ), the calculation of the X -marginal density, namely:

PX ( x, t | x0 , t0 ) := ∫ P( x, u , t | x0 , 0, t0 )du 

(1.9)

can generally not be worked out analytically (for a fully soluble example with multiplicative noise σ( x) see e.g., [1.4]). Due to the absence of microscopic reversibility, (i.e., detailed balance) for the process Eq. (1.7), even the stationary marginal density PX ; s ( x) := lim t →∞PX ( x, t | x0 , t0 ) cannot be exactly obtained in general. For short correlation times, the stationary measure PX ; s ( x) has been calculated by perturbation methods. Although the Ornstein-Uhlenbeck process is commonly invoked, the absence of analytically solvable situations, when using Eq. (1.5) as driving noise, is often considered as a drawback. To explicitly observe how noise correlations influences the dynamics, one may study alternative colored noise sources leading to analytical approaches. Such an alternative is given by the random velocities (RV) jump processes. Basically, RV dynamics lead to motion with finite propagation speeds and approaches, for appropriate scaling limits, either the WGN or White Shot Noise. The simplest process of the RV-class is given by the SDE on  : X = σI , (1.10) t

t

with σ > 0 and the noise source I t is a two-states, continuous time, Markov chain also known as the dichotomous noise or the Telegrapher’s process, (TP). The process I t takes

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values in the set of two velocities {−v, v} and possesses exponentially distributed switching rates connecting both velocities. In the symmetric situation, a single switching parameter λ > 0 fully characterizes the process. Langevin equations with this type of noise are thoroughly studied in [5] and more recently in [6]. The process I t has zero mean, exponentially decaying time correlations of the form

〈 I t , I s 〉 = v 2 e −2 λ (t − s )

t ≥s

(1.11)

and converges in the diffusive limit (i.e., for ε → 0 and when using the re-scaling v  v / ε and λ  λ / ε 2) to WGN. Note that correlations of the TP process Eq. (1.11) coincide with those of the OU process Eq. (1.6). For random evolutions of a scalar state variable X t ∈ , whose dynamics is given by

dX t = [ f ( X t ) + σI t ] dt ,   X 0 = x0 , I 0 = i0 ,

(1.12)

the stationary, marginal probability density PX ; s ( x) associated with Eq. (1.12) can be explicitly derived and reads, ([1.5] section 9.5):   f ( x) −2 λ   dx  f 2 ( x ) − v 2σ 2 ( x )  e

∫



PX ; s ( x) dx =  σ( x)

| f 2 ( x ′ ) − v 2 σ 2 ( x) |

Θ( x)dx,

(1.13)

with  the normalization factor and Θ( x) = I[ x , x ] ( x) stands for indicator function, − + showing that PX ; s has a compact support. The support boundaries x+ and x− are the stable critical points given by the unique solutions to f ( x) + σv = 0 respectively f ( x) − σv = 0 verifying f ′ ( x± ) ± σv < 0. Despite to the fact that dichotomous noise drives the system out-of-equilibrium (i.e., it breaks the detailed balance), Eq. (1.13) offers the exceptional possibility to analytically observe the influence of the correlation parameter λ upon the stochastic dynamics X t . In the limit λ → ∞ with the re-scaling v = λ , the invariant measure Eq. (1.13) converges to the one given in Eq. (1.4). The dynamics given by Eq. (1.12) has been extensively studied, (without being exhaustive see for example [1.4, 1.5, 1.7, 1.8]). The exact expressions for invariant measures Eqs. (1.4) and (1.13) offer important and extensively used tools to explicitly discuss the influence of noise on simple though nonlinear dynamical systems. One may therefore wonder whether other noise sources do offer similar opportunities. This last question can be answered affirmatively and this forms the core motivation of our present contribution in which the noise source dZ t entering Eq.(1.1), while being correlated, still allow explicit derivation of the associated invariant measures. The noise sources to

On Stochastic Processes Driven by Ballistic Noise Sources

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be considered in the sequel are non-Gaussian and exhibit finite time-correlations which in our case is the signature of a ballistic behavior. Two families of such noise sources will be constructed by using the techniques of “lumping Markov processes’’ and their applicationrelevant mathematical properties will be explored. Direct connections of our noise sources with fundamental physics, in particular quantum mechanics will be briefly exposed. One should also realize that noise with ballistic signature are natural. Indeed, for very short timescales, even the random movement of Brownian particles, theoretically respecting the well known square root relation

〈 (Wt − Wt + h ) 2 〉 ∝ h,

(1.14)

has in fact to be replaced by ballistic motion [1.2], characterized by mean displacements proportional to time span:

〈 (Wt − Wt + h ) 2 〉 ∝ h 2 .

(1.15)

Schematically, our paper is organized as follows: in section 2, we introduce two ballistic, non-Gaussian Markovian processes, one parabolic the other hyperbolic, with long range correlations. The construction makes use of the possibility to aggregate certain states of Markov processes and to re-obtain a Markov dynamics for the coarse grained states —a construction also referred, as the Lumpability of Markov processes. In section 3 the non-stationary lumped processes are used as noise generators in SDEs of the type given in (1) and we derive simple relations for the marginal probability measures. We also briefly discuss the stationary partner processes obtained via the Darboux transformation. In the last section several illustrations chosen from off-equilibrium statistical physics, (geometric Brownian motion and coupled phase oscillators) and optimal stochastic control are worked out explicitly.

1.2 Ballistic noise sources with long range correlations 1.2.1 Ballistic Diffusion Process Probably the most simple way to construct a centered ballistic noise Z t with long-range time correlations is to consider a random mixture of two Brownian motions BM (±β) ; one

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with drift β > 0 the other with drift −β. Formally this is written as Z =  ⋅ BM (β) where  is a Bernoulli random variable independent of BM taking values in {±1}. One realization of Z will choose, say with probabilities p (respectively 1 – p), the value 1 (respectively –1) and undergoes thereafter Brownian motion with constant drift β. By lumping the enlarged process X t = ( , BM t (β)) defined on the state space Ω = {−1, +1} ×  appropriately onto  , it is shown in [1.9] (see Example 2) that Z t is a  time homogenous, non-Gaussian Markov diffusion on  , solution to the Langevin equation dZ t = β tanh (βZ t )dt + dWt ,



(1.16)

with initial condition z0 related to the Bernoulli random variable according to Z 0 = z0 = arctanh (2 p − 1) / β. The associated Fokker-Planck equation is: 1 ∂t P( z , t | z0 , 0) = −∂ z β tanh (βz ) P( z , t | z0 , 0)  + ∂ z , z P ( z , t | z0 , 0). 2



(1.17)

The transition kernel of Z t reduces for β = 0 to the Brownian transition density (i.e., Z t = Wt ) and for β > 0 , the solution of Eq. (1.17) reads [1.10]: P ( z , t | z0 , 0) =



e

−

β 2t 2

cosh (βz )

2πt cosh (βz0 )

e

− ( z − z0 )2 2t

.



(1.18)

For z0 = 0 (i.e., if  follows a symmetric Bernoulli distribution with p = 1/ 2 ), Z t is a centered noise source exhibiting moreover the following properties [1.10]:

• The moments read as:

(

)

〈 Z t2 n +1 〉 = 0, 〈 Z t2 n 〉 = ( −t ) ⋅ H 2 n i βt , n = 1, 2,..., n



(1.19)

2

where i = −1 and where H 2 n are the Hermite polynomials.

• Time correlations are given by:

〈 Z t , Z s 〉 = min {s, t } + β 2 st , from which for β ≠ 0 , a long time range of the ballistic type subsists.

(1.20)

• Brownian bridge issues as discussed in [1.11] show that conditioning the Zt process so that Z 0 = ZT = 0 , with T > 0 a fixed time, the resulting conditioned process is the unique Brownian bridge with nonlinear drift term.

In view of the construction of Z t it is immediate to realize that Eq. (1.18) can be rewritten as the superposition of two drifting Gaussians:

−  1 + bβ tanh ( z0 )  1 P( z , t | z0 , 0) = ∑  e  2  2πt b ∈{ ± 1} 

( z − z0 + bβt )2 2t



(1.21)

which is the main reason why we will be able to explicitly compute marginal densities PX ( x, t ) in eq. (1.1) without computing the joint density P ( x, z , t ) first. Note that the timediscrete version of the Z t process is discussed in [1.11] and [1.15].

On Stochastic Processes Driven by Ballistic Noise Sources

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1.2.2  Ballistic Random Velocity Noise To remove the parabolic nature of Eq. (1.16)—implying that probability propagates with infinite velocity—and without losing the explicit solvability, we replace in the above construction the Brownian motion BM (v) with asymmetric dichotomous Markovian noise I t , alternately taking values in {−v, + v} . Switching from v to −v (resp. from −v to v) is exponentially distributed with rate λ + β (resp. λ − β ) where β verifies 0 < β < λ . The parameter β induces asymmetric switching rates between the states ±v resulting in a biased dichotomous noise source with mean β v λ

(1.22)

 β 2  2 −2 λ|t − s| 2 v + . 1 − 2  v e λ2  λ 

(1.23)

〈 It 〉 = −

and correlations:

〈 It , I s 〉 =

β2

Inspired by the construction of the noise source given in eq. (1.16), we now consider the random velocity process Yt given by a random mixture of I t and −I t according to Y = I , Y = y , I = i (1.24) t

t

0

0

0

0

and where the Bernoulli random variable  takes, independently of I t , values in {±1} with equal probabilities. Hence symmetry is restored and if we take averages over trajectories we clearly have 〈Yt 〉 = 0 . For the correlation of the process Eq.(1.24) we find accordingly

〈Yt , Ys 〉 =

 β 2  2 −2 λ|t − s| 2 2 λ v st − 14 , 1 − 2  v e λ2  λ 

β2

(1.25)

exhibiting, via the st contribution, a ballistic behavior. The triple ( , Yt , I t ) is a Markov process with state space Ω = {±1} ×  × {± v}. Let us now show that by lumping the process appropriately onto  × {± v} that, exactly as in the parabolic case above, the resulting statistics is equal in law to (Yt , I t (Yt )) with Yt the solution of the evolution equation

Yt = I t (Yt ),

(1.26)

and where I t (Yt ) stands for a state inhomogeneous TP with state space {−v, + v} and where  β  the sojourn times in the states ±v are Yt -dependent with rates  λ ± β tanh  Yt   . In  v   view of the construction of Yt we will be able to explicitly compute marginal densities P( y, t | y0 , 0) (for normal boundary conditions and symmetric initial conditions) using the relation

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β P( y, t | y0 , 0) = cosh  v

 y  T ( y, t | y0 , 0) 

(1.27)

where the field T ( y, t | y0 , t0 ) solves a dissipative wave equation (i.e., the Telegrapher’s equation) of the form:

∂t ,t + 2λ∂t + β 2  T ( y, t | y0 , t0 ) = v 2 ∂ y , y T ( y, t | y0 , t0 )  

(1.28)

and whose well known solutions can be given explicitly in terms of Bessel functions, ([1.22, 1.23], and see also Eq. (1.71) below). Let us first focus on the Master equation for Eq. (1.26) and write for short P( k , j ) ( y, t ) := P( y, k , t | y0 , j , t0 ) for the conditional probability density to observe Yt in the interval [ y, y + dy ] with noise state I t = k knowing that at time t0 we had Yt = y0 , and 0 I t = j. The conditional densities evolve according to the Master equation: 0



  β  β ∂t − v∂ y + [λ + β tanh  y   P( + v , j ) =  λ − β tanh  v v      β  β ∂t + v∂ y + [λ − β tanh  y   P( − v , j ) =  λ + β tanh  v v   

 y   P( − v , j ) , 

(1.29)

 y   P( + v , j ) , 

where P( ± v , j ) stands for P( ± v , j ) ( y, t ) . We suppose natural boundary conditions and symmetric initial conditions:

P( + v , j ) ( y, t0 ) = P( y, + v, t0 | y0 , j , t0 ) =

λ − β tanh (βvy ) δ ( y0 )δ ( j − v), 2λ



P( − v , j ) ( y, t0 ) = P( y, − v, t0 | y0 , j , t0 ) =

λ + β tanh (βvy ) δ ( y0 )δ ( j + v). 2λ

One verifies that for j ∈{−v, + v} in Eq. (1.29), both TPD’s P( ± v , j ) solve the spaceinhomogeneous, second order hyperbolic equation:

  β ∂t ,t + 2λ∂t + 2vβ ∂ y  tanh  v  

  y   − v 2 ∂ y , y  P( ± v , j ) = 0.  

(1.30)

Integrating out j and using Eq. (1.27) the relation with Eq. (1.28) is readily established. We now turn to the transition kernel b ( x, i, t | y, j , s ) of the triple ( , Yt , I t ) where x, y ∈, i, j ∈{−v, v}, b ∈{−1,1}, 0 < s < t. Fix velocity i0 ∈{−v, v} and location x0 at initial time t0 = 0 and write for short

u+↑ ( x, t | 0) := −1 (Yt = x, I t = +1| Y0 = x0 , I 0 = i0 , 0),

(1.31)



u−↑ ( x, t | 0) := −1 (Yt = x, I t = −1| Y0 = x0 , I 0 = i0 , 0),

(1.32)

↑

for the transition kernel of (Yt , I t ) := ( −1, Yt , I t ) and similarly

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On Stochastic Processes Driven by Ballistic Noise Sources



u+↓ ( x, t | 0) := 1 (Yt = x, I t = +1| Y0 = x0 , I 0 = i0 , 0),

(1.33)



u−↓ ( x, t | 0) := 1 (Yt = x, I t = −1| Y0 = x0 , I 0 = i0 , 0).

(1.34)

↓

for the transition kernel of (Yt , I t ) := (1, Yt , I t ) . The time evolution of Eqs. (1.31) and (1.32) u+↓ and u−↓ is governed by the equations:

[∂t + v∂ x + (λ + β)]u+↓ ( x, t | 0) = (λ − β)u−↓ ( x, t | 0),



[∂t − v∂ x + (λ − β)]u−↓ ( x, t | 0) = (λ + β)u+↓ ( x, t | 0),

and analogously for u+↑ and u−↑ [∂t + v∂ x + (λ − β)]u+↑ ( x, t | 0) = (λ + β)u−↑ ( x, t | 0),

[∂t − v∂ x + (λ + β)]u−↑ ( x, t | 0) = (λ − β)u+↑ ( x, t | 0).

Separating the fields u+↑ ( x, t | 0) and u−↑ ( x, t | 0) by differentiating the above equations with respect to t and x , it is immediate to obtain that both conditional densities satisfy the same dissipative wave equation: ∂t2 φ↑ ( x, t ) + 2λ∂t φ↑ ( x, t ) = v 2 ∂ 2x φ( x, t ) + 2βv



∂ ↑ φ ( x, t ). ∂x

(1.35)

where φ↑ ( x, t ) stands for either u+↑ ( x, t | 0) or u−↑ ( x, t | 0) . Analogously the fields u+↓ ( x, t | 0) and u−↓ ( x, t | 0) both satisfy : ∂t2 φ↓ ( x, t ) + 2λ∂t φ↓ ( x, t ) = v 2 ∂ 2x φ( x, t ) − 2βv



∂ ↓ φ ( x, t ). ∂x

(1.36)

Introducing the transformations:

T ( x, ± v, t | x0 , i0 , 0) = e + βvx φ↑ ( x, t ) = e + βvx u±↑ ( x, t | 0),

(1.37)



T ( x, ± v, t | x0 , i0 , 0) = e −βvx φ↓ ( x, t ) = e + βvx u±↓ ( x, t | 0),

(1.38)

we observe by an elementary verification that functions T ( x, k , t | 0) , k ∈{± v} solve Eq. (1.28). Accordingly, and in full similarity with section 2.1, the non-homogeneous process Eq. (1.26) with master equation Eq. (1.29) can be viewed as a lumped version of ( , Yt , I t ).

1.2.3 Connections with Quantum Mechanics The ballistic noise sources given by Eqs. (1.16) and (1.24) with their respective FokkerPlanck equation (1.17) and (1.30) are related in several aspects to quantum mechanics (QM) and this is now shortly reviewed here.

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1.2.3.1  Quantum measurement process Measurements processes on a quantum system Q often use a classical device C. Hence, a consistent modeling of QM measurements requires an ad-hoc modeling of the Q – C coupling. Any measurement, can be viewed as a transfer of information occurring from Q to C and thus leading to an irreversible process. Mathematically speaking this dissipative structure can be described by a semigroup of trace preserving maps on the whole Q – C system [1.12]. Underlying this measurement dynamics, one may naturally introduce stochastic processes, (discrete piecewise deterministic processes in [1.12, 1.13] and diffusion processes in [1.14] ) and their associated master equations. Let us consider here, following [1.14], the generalized nonlinear Schrödinger equation for the quantum state ψ(t ) in a relevant Hilbert space:

ψ (t ) = −iH ψ (t ) + Kt [〈 H 〉 Ψ(t ) − H ] ψ (t )

(1.39)

   dissipative term

where H is the Hamiltonian operator. In Gisin’s view, the dissipative contribution in Eq. (1.39) assimilates the action of a measurement apparatus to an interaction with a finite temperature heat bath with fluctuations given by a diffusion process Kt :

dKt = f ( Kt .ψ t )dt + dWt ,

(1.40)

where dWt is the WGN and the underlying stochastic integral now appearing in Eq. (1.39) is interpreted in the Stratonovich sense (which is another mathematical interpretation of the stochastic integral ∫σ( s, X s )dWs , see e.g. [1.1], section 4.3.6). Observe that the nature of the Q – C interactions here implies that the heat bath itself evolves with the state of the whole system. This is taken into account by allowing an explicit ψ -dependent drift f ( K t .ψ t ) in Eq.(1.40). For a spin system, the formalism summarized in Eqs. (1.39) and (1.40) implies:

d ψ t = −iω Pψ t dt + [〈 P 〉 ψ − P ] ψ t  dkt , dkt = f (kt , ψ t ) dt + dWt , t

(1.41)

where the operator P has two eigenvalues (the spin states), ω is a (Larmor) frequency and the  makes clear that we do stochastic integration according to Stratonovich’s definition. One further requires the average pt = 〈 P 〉 ψ to be constant in time; this to ensure a cont sistent description of measurements overlapping in time. After calculations, see [1.14], this imposes the choice f (kt , ψ t ) = 1 − 2 pt and therefore:



2 pt (1 − pt )dWt ,  dpt =  dp = 2 p (1 − p ) (2 p − 1)dt + dW  t t  t t  t

(Ito), (Stratonovich).



(1.42)

Now writing tanh( Z t ) = (2 pt − 1) , one immediately verifies that Eq.(1.42) coincides with Eq.(1.16) thus showing how our ballistic diffusive noise source enters naturally into QM measurement modeling.

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On Stochastic Processes Driven by Ballistic Noise Sources

1.2.3.2   Quantum random walk The classical random walk Rt on the integer lattice  is commonly described by probability densities P(n, t ) := Prob {Rt = n | R0 = 0} . In discrete time, the associated master equation reads: P(n, t + 1) = g n +1 P(n + 1, t ) + rn −1 P(n − 1, t ), n ∈ , t ∈  + ,



(1.43)

where rn and g n are left respectively right transition rates characterizing the walker. For quantum random walk (QRW), one defines a random evolution on a spinor of wave functions Ψ L (n, t ) and Ψ R (n, t ) with the rule:  Ψ L (n, t + 1) 1  1 1   Ψ L (n + 1, t )  Ψ (n, t + 1) = 2  1 −1  Ψ (n − 1, t ) R R



(1.44)

and PQRW (n, t ) = [Ψ L (n + 1, t ) + Ψ R (n + 1, t )]2 gives the probability to find the QRW at position n at time t . It can be seen that the interferences present in the QRW produce asymptotically in time a ballistic term of the form 〈 Z 2 (t )〉 ∝ t 2 which precisely follows Eq.(1.19) for n = 1, (i.e., 〈 Z 2 (t )〉 = t + β 2 t 2 ). The QRW propagates therefore quadratically faster than the classical random walk for which we have 〈 Rt2 〉 ∝ t . It is shown in [1.16] that, in the appropriate continuous limit, the dynamic behavior of the QRW closely matches the process given in Eq.(1.16).

1.2.3.3  Dirac and Klein-Gordon equations As pointed out in [1.17] and more recently in [1.18], a close correspondence exists between the telegrapher’s and the Dirac equation in (1 + 1)-dimension. Writing the Dirac equation in its canonical form, we have:  µ m0 v 2  −1 i γ ∂ µ −  ψ = 0, with ∂ µ := v ∂t , ∂ x , ψ := ( ψ1 , ψ 2 ) ,   

(



)

(1.45)

with i = −1 and the Weyl’ representation:

 0 1 γ0 =   1 0

and

 0 −1 γ1 =  .  1 0 

Defining ϕ1 and ϕ 2 via the relations

ψ1,2 ( x, t ) = e − λt ϕ1,2 ( x, t )

(1.46)

one verifies that the evolution for ϕ1,2 coincides with the telegrapher’s Eq.(1.28) provided m v2 one identifies 0 with λ 1 − [β 2 / λ 2 ] , (remember from the definition of the model that i β < λ ). The i = −1 factor is reminiscent of the analytic continuation needed to convert diffusion-like processes to Schrödinger-like equations. Accordingly, this formal analogy with Eq.(1.28) allows to view a Dirac particle in one dimension as moving back and forth

12

Contemporary Topics In Mathmatics and Statistic Applications

on  at the speed of light v with random reversal of directions, (i.e., a Zitterbewegung), [1.18] [1.19]. Similarly, introducing φ1 and φ 2 , defined via the relations

ϕ1,2 ( x, t ) = e

 xβ  −λ t ±   vλ 

φ1,2 ( x, t ),



(1.47)

into the biased evolution described by Eqs.(1.35) and (1.36), one shows that φ1,2 ( x, t ) again obey Eq.(45) provided the same identification m0 v 2 i = λ 1 − [ Rt2 ] is made. One recognizes a Lorentz type dilatation factor. In [1.18], it is shown that the biased telegrapher’s Eqs.(1.35) and (1.36) are interpretable as unbiased telegrapher’s processes viewed from a Lorentz transformed frame with boost velocities being the mean drift velocities induced by the left and right biases.

1.3 Langevin equations driven by lumped processes The lumped processes Yt and Z t introduced in section 2 will now be used as driving noise sources in scalar random evolutions. We shall namely consider: X t = f ( X t ) + g ( X t )Yt



and

dX t = f ( X t )dt + g ( X t )dZ t ,

(1.48)

with f bounded and Lipschitz and where the strictly positive diffusion coefficient g (x) satisfies

∫ g ( x )

−1

dx < ∞ . Due to the representation of Yt and Z t as lumped processes, we

immediately have:

• the marginal stationary probability density resulting from the Yt noise source



PX(Y; s) ( x) =

  g ( x) f ( z) exp  −2λ ∫   | f + ( x) f − ( x) | f + ( z ) f − ( z )dz   x   g ( z)   dz  Θ( x), × cosh  2βv λ ∫   f + ( z ) f − ( z )   



(1.49)

with the notation f ± ( x) = f ( x) ± vg ( x) . To establish Eq.(1.49), one observes that

Lumpability implies that PX(Y; s) ( x) can be written as a sum of two stationary measures:

1 (Y , − ) 1 PX ; s ( x) + PX(Y; s, + ) ( x), 2 2

(1.50)

with PX(Y; s, ± ) ( x) being associated with the processes



PX(Y; s) ( x) =

X t = f ( X t ) + g ( X t )Yt ±

(1.51)

where Yt ± are two symmetrically biased dichotomous noise sources with respective average ± βv , (see Eq.(22)). Accordingly, (see Eq.(1.50) in [1.5]), the λ

On Stochastic Processes Driven by Ballistic Noise Sources

13

stationary measures associated with Eq.(1.51) read: (Y , ± ) PX ; s ( x) =

x  g ( x) f ( z ) ± βvλg ( z )  exp  −2λ ∫ dz  Θ ± ( x)  ± | f + ( x) f − ( x) | f+ ( z) f− ( z)  

(52)

• the marginal stationary probability density resulting from the Zt noise source. Interpreting the underlying stochastic integrals according to the Stratonovich definition:



PX( Z; s) ( x) =

 x dz   x 2 f ( z )  1 cosh  2β∫ 2  ⋅ exp ∫ 2 dz   g ( x) g ( z )    g ( z ) 

(1.53)

Here again, using Lumpability, one can write PX( Z; s) ( x) = PX( Z; s, − ) ( x) + PX( Z; s, + ) ( x)

with

PX( Z; s, ± ) ( x) =

 x 2 ( f ( z ) ± β )  1 exp ∫ dz  . 2  g ( x) g ( z )  

(1.54) (1.55)

In both cases Eqs.(1.49) and (1.53), we may easily calculate the position of the probability modes, (intuitively the most probable locations x* ) which follows from:

d ( ν) PX ; s ( x) | * = 0 x= x dx

with

ν ∈{Y , Z }

(1.56)

which for Eq.(1.49) implies:

[ − g ′ ( x * ) / g ( x * ) ] f 2 ( x* ) − v 2 g ( x * ) g ′ ( x * ) + 2 f ( x * ) f ′ ( x * )



x   g ( z)   + 2λf ( x* ) + 2βvg ( x* ) tanh  2βv ∫  dz  = 0  f + ( z ) f − ( z )   

(1.57)

and for Eq.(1.53), we have:

 x* dz  − g ( x* ) g ′ ( x* ) + 2 f ( x* ) + 2β tanh  2β ∫ 2  = 0. g ( z )  

(1.58)

Apart from the specific nature of the drift f ( x) , the positions  x* will depend on noise control parameters like β, λ and hence on the correlations of the noise sources. New emerging features (i.e., noise induced extra modes of PX( ν; s) ) are to be expected and this even in presence of additive noise sources. For SDE driven by WGN, such phenomena does occur only in presence of multiplicative noise sources, i.e., when noise amplitude itself depends on the state variables X t . For antisymmetric drift fields, f ( x) = − f ( − x) , and symmetric diffusion coefficient, g ( x) = g ( − x), leading to symmetric marginal stationary densities PX( ν; s) ( x) = PX( ν; s) ( − x) , the signature of noise induced transitions is given by the sign of the curvature of stationary probability densities at the origin. We obtain:

14

Contemporary Topics In Mathmatics and Statistic Applications ^2



and

: B ¨   · ¸ © ¨ 2 f a 2 (0) · ¸ © ©B  ¸ © g aa (0) 2v 2 ¹ ¸ L d 2 (Y ) ª ¸ Ps ,Y (0) ©  2 2 f a (0)  4 PX ; s ( x) |x = 0 = ¸ © g (0) dx 2 v g (0) 2 ¸ © ¸ © ¸ © ¹ ª

(1.59)

 g ′′ (0) β2  (Z ) P x f ( ) | = − + 2 (0) + 4 ′  Ps , Z (0). X ,s x =0  dx 2 g (0) 2   g (0)

(1.60)

d2

One observes that when λ / v 2 = 1 , the rescaling λ / ε 2 and v / ε for ε → 0 ensures that Eq.(1.59) consistently reduces to Eq.(1.60). From Eqs.(1.59) and (1.60), one therefore observes that the Z-noise with ballistic parameter β acts similarly as the Y-noise with an enhanced ballistic contribution βˆ 2 = β 2 + f ′ 2 (0)/2v 2 . In particular, for additive noise with g ( x) constant or for convex diffusion coefficient g ′′ (0) > 0 and attractive drift f ′ (0) < 0, we conclude that the ballistic contribution can change the sign of the curvatures thus creating noise induced transitions [1.5].

1.3.1 Related Stationary Processes Due to their repulsive drifts, both the Yt and Z t noise sources are non recurrent stochastic processes, (i.e., stationary regimes do not exist). Stationary noise sources related to Yt and Z t can be obtained in several ways and two simple and natural ways are now briefly exposed for completeness. 1. Consider linearly damped processes driven by Yt or Z t and use the resulting generalized Ornstein-Uhlenbeck (OU) process as a stationary noise source. 2. Reverse the drift sign, (thus creating an attractive drift), in Eqs.(1.16) and (1.24)

1.3.1.1  Stationary processes resulting from OU processes (i) For the Z t case, this leads to rewrite instead of Eq.(1.7), the new dynamical system:



dX t = f ( X t , t )dt + dZ t , X 0 = x0 , dZ = Udt ,  t  U 0 = 0, dU t = − λU t dt + σdVt , dVt = β tanh(βVt ) + dWt .

(1.61)

In Eq.(1.61), the noise source Z t is now a non-Gaussian stationary process, which, by using the Lumpability property for the (U t , Vt ) processes in Eq.(1.61), enables to write the marginal TPD:

15

On Stochastic Processes Driven by Ballistic Noise Sources

1 G ( − mt , σt2 ) + G (mt , σt2 )  , (1.62) 2 where G (m, σ 2 ) stands for the normal density with (time-dependent) mean mt and 2 variance σt given by:





PU (u , t | 0,0) = ∫ P(u , v, t | 0,0,0)dv = 

mt =

σβ (1 − e − λt ) λ

and

σt2 = σ 2 (1 − e −2 λt ).

(1.63)

(ii) For the Yt noise source, one would write:

dX t = f ( X t , t )dt + U t ,  dU t = − λU t dt + σI t ,

X 0 = x0 , U 0 = 0,



(1.64)

with I t being the process defined in Eq. (1.24). Again, Lumpability offers the possibility to write:

PU (u , t | 0, 0) = ∫ P(u , v, t | 0, 0, 0)dv = 12  − (t ) +  + (t ) , 

(1.65)

where  ± (t ) are two symmetrically biased TPD’s which, despite to the underlying linearity of the U t process in Eq.(1.64), lead to complex expressions, (see [1.20]). Summarizing, as far as non-Gaussian stationary noise sources are concerned, only the scheme expressed by Eq.(1.61) will provide us with a noise source leading to further relatively simple calculations.

1.3.1.2  Stationary processes derived via Darboux transformation We may consider the naturally associated processes obtained by reversing the drifts. In this case Eqs.(1.30) and (1.7) respectively read: and

[∂t ,t + 2λ∂t − 2β ∂ y [ tanh (βy )] − ∂ y , y ]( ± , j ) ( y, t ) = 0

1 ∂ z , z  ( z , t ), 2 where, in Eq. (1.66), we have taken without lost of generality v = 1 .

∂t  ( z , t ) = ∂ z [β tanh (βz ) ( z , t ) ] +

(1.66) (1.67)

We introduce the Darboux integral transform Dx [1.21]:

x F ( x, t ) :=

e

−

cosh

β 2t 2 2

x

F (ξ, t ) cosh (βξ) d ξ, (βx) ∫−∞

(1.68)

for functions F ( x, t ) for which the above integral is finite. For initial conditions lim t →0P( y, t ) = δ ( y ) and lim t →0P( z , t ) = δ ( z ) , a direct calculation using Eq.(1.68) shows that Dy [∂ y P ( y, t )] and Dz [∂ z P( z , t )] solve Eqs.(1.66) and (1.67) provided that Py ( y, t ) := ∂ y P( y, t ) and Pz ( z , t ) := ∂ z P( z , t ) respectively solve the telegrapher’s and the heat equation:

16

Contemporary Topics In Mathmatics and Statistic Applications

[∂t ,t + 2λ∂t ]Py ( y, t ) = ∂ y , y Py ( y, t ), with



Py ( y, 0) = δ ( y ),

(1.69)

and ∂t Pz ( z , t ) = 12∂ z , z Pz ( z , t ), with



Pz ( z , 0) = δ ( z ).

(1.70)

For Eq.(1.69), we have: Py ( y, t ) =



∞

where I 0 ( z ) = ∑ n =0

{(

e − λ 2t [1 + λ∂t ] I 0 2

}

)

t 2 − y 2 [Θ ( y + t ) − Θ ( y − t ) ] ,

(1.71)

( z / 2) 2 n

is the modified Bessel function and Θ(⋅) stands for the (n!) 2 Heaviside function. We hence have:

 ( y, t ) =

Py ( y, t ) cosh (βy )

+ β∫

y −∞

Py (ξ, t ) sinh (βξ)d ξ −1

For the heat equation Eq.(1.70), using Pz ( z , t ) = ( 2πt ) e

P ( z , t ) = e −β

2 2t

cosh (βz ) 2πte − z

2 2t

+ β2 π cosh 2(βz )∫

−

z2 2t

(1.72)

, one obtains:

zβ 2t + β t /2 −ξ 2 zβ 2t −β t /2

e

d ξ, (1.73)

which can also be found in [1.24]. From Eq.(1.73), one immediately observes that β cosh −2(βz )dz . Moreover, the asymptotic expansion of the Bessel 2 function for large times t  λ −1 yields Py ( y, t )  Pz ( z , t ) which, as shown in [1.22], implies that P( y, t )  P( z , t ) in the asymptotic regime. Note that for arbitrary initial conditions, the general solutions to Eqs.(1.66) and (1.67) can be developed over complete sets of relevant eigenfunctions of the associated Sturm-Liouville problem. This has been explicitly worked out in [1.24, 1.25, 1.26].

lim t →∞ P( z , t )dz =

As far as noise sources are concerned however, neither Eq. (1.66) nor Eq.(1.67) seem to provide stationary noise sources that will lead to relatively simple analytic approaches.

1.4  Illustrations involving the ballistic noise sources 1.4.1 Generalized geometric Brownian motion As illustrations for the driving action of the Yt and Z t noise sources introduced in section 3, we briefly consider SDEs structurally related to geometric Brownian motion dynamics finding various applications in finance, population growth- and tumor dynamics.

1.4.1.1  Multiplicative noise using the Yt -noise source Consider the multiplicative noise Langevin equation

On Stochastic Processes Driven by Ballistic Noise Sources

X (t ) = ρ + Yt  X (t )



17 (1.74)

with ρ ∈ a constant drift and with Yt as defined by Eq.(1.24). For fixed initial condition X (t = 0) = x0 > 0 we immediately find: t



X (t ) = x0

∫ ρ+Ys  ds . e 0

(1.75)

Note that the use of a Telegrapher’s process I t in Eq.(1.74) has been thoroughly discussed in [1.27]. Following [1.28] we now briefly discuss two related asymptotic stability issues for the solution X (t ) of Eq.(1.74).

• Path-wise asymptotic stability. As explicitly shown in section 2.2, the Yt noise source is the superposition of two symmetrically biased Telegraphic processes Ys↑,↓ (see eqs. (1.31-1.34)) with drifts ±vβλ . Thanks to ergodicity, we can write: 1 t ↑ ,↓ Ys ds t ∫0



t →∞

 → 〈Yt↑,↓ 〉 = ± vβλ.

(1.76)

Imposing the trajectories of Eq.(1.74) to be stable for both Bernoulli-states {±v} , we conclude that pathwise stability will follow provided ρ + vβλ < 0 .

• Asymptotic stability in the mean. For stability in the mean, one requires the real part

of the eigenvalues of the infinitesimal generators M ± to be negative, (see e.g.,[1.28]). We have  [−λ − β + ρ ± v] M± =  λ −β 



 λ+β , [ − λ + β + ρ  v ]

with eigenvalues Γ + = { A + B+ , A − B+ } and Γ − = { A + B− , A − B− } where:

A = ρ − λ, and B± = λ 2 + v 2  2βv .

(1.77)

Using 0 < β ≤ λ and completing squares in B± , it is easily established that max(Γ + ∪ Γ − ) = A + B− < ρ + v . For stability in the mean, one requires therefore ρ + v < 0 β which clearly differs from the condition ρ + v < 0 required for the path-wise stability. In λ fact, using elementary algebraic manipulations, it is seen that stability in the mean implies path-wise stability if λ > v and that path-wise stability implies stability in the mean if λ 0 we can write: t



X (t ) = x0

∫ ρ+ Zs  ds . e 0

(1.79)

Note that for ρ > 0 and the use of a WGN instead of Z t in Eq.(1.78) we exactly find the Black and Scholes financial model. As far as the Z t noise source is concerned, the Lumpability property together with a logarithmic change of variable, allow to immediately write the TPD in the compact form for x ∈  + :



 [ln ( x ) − (ρ −β )t − ln ( x0 ) ]2 [ln ( x ) − ( ρ+ β))t − ln ( x0 ) ]2  1 −  1 − t 2t 2 + e   e 2 2    dx P( x, t | x0 , 0) dx =  (1.80) x 2πt β 2t 2

 ln  xx  − ρt  2      0         x e   −  2t cosh β ln   − ρt    e =  dx. x 2πt          x0    −

The time asymptotic regime of Eq.(1.80) offers different behaviors as a function of the ballistic contribution. We indeed have:



  (i ) δ ( x) for (ρ − β) < 0 and (ρ + β) < 0 ,  1 1  δ ( x) + G(ρ−β ) ( x, t ) for (ρ − β) < 0, lim P( x, t | x0 , 0)   (ii ) 2 2 t →∞  1 1  (1.81) (iii ) 2 Gρ+β ( x, t ) + 2 G(ρ−β ) ( x, t ), for (ρ − β) > 0.

where we have written

Gq ( x, t ) =

1 x 2πt

e

−

( ln x − qt )2 2t

.

The case (ii) in Eq.(1.80) offers an interesting perspective for finance. Indeed when ρ > 0 , Eq.(1.74) describes the evolution of an asset with a fluctuating rate ρ + Z t . In this case, strong ballistic contributions (arising when ρ − β < 0 ), drives part of the fortune to the bankrupt 0-boundary for asymptotic times. This is a novel feature compared to the standard Black and Scholes’ dynamics.

1.4.2 Optimal Control Inspired by the work of B. Kappen [1.29, 1.30], we show, using stochastic optimal control theory, that the pinning of the process defined by Eq.(1.16) at some fixed point in time

19

On Stochastic Processes Driven by Ballistic Noise Sources

t f > ti ≥ 0 , essentially coincides with a Gaussian bridge. Consider the stochastic optimal control problem characterized by:

dX τ = u (τ ) + β tanh (βX τ )  d τ + σdWτ , τ ∈ ti , t f        X = x , i  ti

(1.82)

where u(τ) is a control function. To Eq.(1.82), we associate a quadratic cost functional:

 u ( s)2  t C ( xi , ti , u (⋅) ) = φ( X (t f ) + ∫ f ds  + V ( X ( s ), s )  2 ti  2σ 



(1.83)

xi

where the average is taken over all realizations starting at xi. The Hamilton-Bellman-Jacobi (HBJ) equation that can be associated to Eq.(1.82) reads as:

 u 2  σ2 −∂t J ( x, t ) = min  2 + V + (β tanh (βx) + u ) ∂ x J ( x, t ) + ∂ x , x J ( x, t ) (1.84) 2 u   2σ 

where J ( x, t ) := min C ( xi , ti , u (⋅) ) is the optimal cost-to-go (or value-) function. Minimization with respect to u leads to: u = −σ 2 ∂ x J ( x, t ),



(1.85)

leading for J ( x, t ) to the nonlinear PDE:

−∂t J ( x, t ) = V −

σ2 σ2 [∂ x J ( x, t )]2 + [β tanh (βx)] ∂ x J ( x, t ) + ∂ x , x J ( x, t ). (1.86) 2 2

Eq.(1.86) is linearized via the logarithmic transformation: J ( x, t ) = − ln [ Ψ( x, t ) ] ,



(1.87)

leading to a PDE with final condition at t f :



   σ2 ∂ x , x  Ψ( x , t ) ∂t Ψ( x, t ) = V − β tanh (βx)∂ x − 2     − φ( x ) . Ψ( x, t f ) = e

(1.88)

The solution of Eq.(1.88) can be written as:

Ψ( x, t ) = ∫ dyρ( y, t f | x, t )Ψ( y, t f ) = ∫ dyρ( y, t f | x, t )e − φ( y ) 



(1.89)

where ρ( y, t | x, ti ) is the TPD of the diffusion process with killing rate V :

dX t = β tanh (βX t ) + σdWt   X t = X t + dX t with probability 1 − V ( X t , t )dt   X = killed with probability V ( X , t )dt. t  t

(1.90)

20

Contemporary Topics In Mathmatics and Statistic Applications

The path integral formulation for the TPD ρ( y, t f | x, ti ) reads, (see also [1.31, 1.32]): ρ( y, t f | x, t ) = ∫



 − S path  X (t  t f  e all paths

 ) 



(1.91)

with the definition:

S path  X (t  t )  : =  f  

  x (τ) − β tanh (βx)  2  β2 + V ( x , τ ) d τ   . (1.92)  + 2σ cosh 2(βx)    

tf

∫t

Focusing on situations for which σ ≥ 1 an choosing the killing rate as: V ( x) = β 2 1 − σ −2  tanh (βx) 2



(1.93)

Eq.(1.91) to (1.93) imply for t > ti ,

ρ( y, t | x, ti ) =

− β2 2(t − ti )

 cosh ( y )    2πσ (t − ti )  cosh ( x)  e

σ −2

2

−

e

( y − x )2 2 σ 2 (t − ti )

.



(1.94)

Note that Eq.(1.93) implies V = 0 for σ = 1 and Eq.(1.94) coincides with Eq.(1.18) in this case. A control u in Eq.(1.82) that pins the process to x = 0 at time t f (i.e., considering the stochastic bridge from ti to t f ), is realized if we choose a final cost function of the form φ( x) = δ ( x), δ being the Dirac measure. The use of Eq.(1.89) and (1.94) imply:



with

2

Ψ( x, t ) = ∫ dy ρ ( y, t f | x, t )δ ( y ) = N (t f − t ) cosh (βx) − σ e

−

x2 2 2σ (t f − t )



 (t f − t ) =

e

,

(1.95)

− β2 2(t f − t )

2πσ 2 (t f − t )

which in view of Eqs.(1.85) and (1.87) yields the optimal drift u as: x u = ∂ x ln [ Ψ( x, t ) ] = − 2 − βσ 2 tanh (βx). σ (t f − t )

(1.96)

Introducing the control Eq.(1.96) into the original problem Eq.(1.82), one obtains a final diffusion process:



  X dX τ =  − 2 τ + β(1 − σ 2 ) tanh (βX τ )  d τ + σdWτ  σ (t f − τ ) 

(1.97)

As pointed out in [1.11], it is truly remarkable that for σ = 1, which in the unpinned case corresponds to Eq.(1.16), the resulting nonlinear process coincides with the (Gaussian) Brownian bridge.

21

On Stochastic Processes Driven by Ballistic Noise Sources

1.4.3  Kuramoto Sakaguchi Dynamics Here, we apply the ballistic noise source Z t to the cooperative dynamics of an assembly of globally coupled phase oscillators. In this context, one of the most successful models is doubtlessly the Kuramoto-Sakaguchi (KS) model [1.33], [1.34]. It consists of a population of N coupled phase oscillators where the phase of the k-th oscillator, denoted by θ k , evolves in time according to

d θ k (t ) = ω k dt +

K N

N

∑ sin (θ j − θk ) dt + j =1

2 DdWk (t ),

k = 1, 2,...N .

(1.98)

Here, K is an all-to-all coupling constant and ω k is the natural frequency of oscillator k which is drawn at random from some probability distribution g(ω ) . The random environment is additive WGN, 2 D dWk (t ) , with noise strength D > 0 . When all oscillators share a common frequency ω , the dynamics, in a ω -rotating framework can be reduced to: by

d θ k (t ) =

K N

N

∑ sin (θ j − θk ) dt + j =1

2 DdWk (t ),

k = 1, 2,...N .

(1.99)

The model Eq.(98) can be studied in terms of a complex order parameter eiΘ given



 exp (iΘ ) :=

1 N

N

∑e j =1

iθ j

.

(1.100)

The amplitude  ∈[0,1] measures the phase coherence of the oscillators and Θ represents an average phase of the system. Thanks to , we can identify the cooperative state of the oscillators assembly, namely  = 1 indicates a fully synchronized motion,  = 0 characterizes a fully incoherent behavior and when 0 <  < 1, the assembly possesses a partially synchronized state. After the pioneering works [1.33] and [1.34], the behavior of Eq.(1.98) has been further analyzed in presence of Gaussian noise sources dWk (t ) having white or colored spectral densities, [1.35]-[1.43]. In the limit N → ∞, it can be shown that, tuned by the coupling strength K, a second order type phase transition occurs between the fully incoherent and a partially synchronized state at a critical bifurcation value K c. The explicit dependence of K c on the underlying control parameters, (noise amplitude D, noise coloration,...), has been derived analytically for several Gaussian noise sources and a rather large class of frequency distributions g(ω ) . These analytical and numerical studies underline the following intuitive picture: an increase of the noise correlations decreases the value of K c. The injection of non-Gaussian noise into the dynamics given by Eq.(1.98) will generally preclude analytical discussion and hence do enforce numerical studies to be performed. This numerical approach has recently been adopted in [1.44] for a class of stationary processes for which the invariant measure exhibits a symmetric non-Gaussian, uni-modal probability density. The authors report that the global qualitative picture of the phase transition is preserved. However, the bifurcation value K c explicitly depends on

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Contemporary Topics In Mathmatics and Statistic Applications

the non-Gaussian character of the noise. Adopting a similar research direction, we now analyze the behavior of Eq.(1.98) in presence of the super-diffusive, non-Gaussian noise source given in Eq.(1.16). The β -controlled non-Gaussian character, produces a surprisingly enriched phase transition diagram which include regimes with time oscillations of the order parameter  . This stable temporal modulation is entirely due to the non-Gaussian character of the driving noise tuned by β . For the limiting case β = 0 , the noise model coincides with the standard WGN case. Note that contrary to [1.44], our noise process is non-stationary and exhibits a bi-modal transition probability density.

1.4.3.1  Kuramoto-Sakaguchi model driven ballistic noise In the following, we consider the KS like-dynamics given in Eq.(1.99) where the dWk (t ) are replaced with dZ k (t ) given in Eq.(1.16). Using the representation based on a Bernoulli superposition explained in section 2, it is straightforward to see that Eq.(1.98) can be effectively rewritten as:

d θ k (t ) = ω k dt +

K N

N

∑ sin (θ j − θk )dt + j =1

2 DdWk (t ),

k = 1, 2,...N , (1.101)

where the frequencies ω k are now drawn randomly from a probability distribution g(ω ) : 1 1 δ (ω − β 2 D ) + δ (ω + β 2 D ) 2 2 Hence we can summarize this observation by saying:



g (ω ) =

(1.102)

The effective action of the non-Gaussian noise sources on the homogeneous frequency model Eq.(1.99) is equivalent to drive an heterogenous bi-frequency KS model Eq.(1.101) with Gaussian noise sources. The stronger the non-Gaussian character, (i.e., the stronger β ), the larger the variance of the distribution g(ω ) in Eq.(1.102). The resulting dynamics defined by (1.101) and (1.102) is thoroughly discussed in e.g., [1.38]-[1.41]. It exhibits the following features: D , a phase transition between a 2 fully incoherent state to a partially synchronized state occurs at a bifurcation value K c (β, D) = 2 D + 4β 2. While the bifurcation value K c (β, D) is β -dependent, the resulting bifurcation diagram topologically coincides with the Gaussian β = 0 case (cf. Fig. 1.1, in Ref. [1.40]).

(a) Close to Gaussian noise regime. When 0 ≤ β


D entirely new dynamic features 2

emerge and a sketch of the (conjectured) global bifurcation diagram is given in Figure 5 in Ref. [1.40]. In summary, one finds β and D dependent critical values K1 (β, D) > 4 D and K 2 (β, D) < K1 (β, D) such that:

23

On Stochastic Processes Driven by Ballistic Noise Sources

• For 4 D ≤ K < K1 (β, D), temporal oscillations of the OP occur and the zero value

(t ) = 0 is repeatedly attained. Both the oscillations amplitude and their period increase with K . Due to the symmetry of g(ω ) in Eq.(1.102), the global average phase Θ in Eq.(1.100) is time-independent during partial synchronization with  > 0 . Its value depends on the initial angle distributions [1.36], [1.40]. Hence, the value of the average global phase can be modified at times when full incoherence, (t ) = 0 , is reached.

• For K1 (β, D) ≤ K < K 2 (β, D) and depending on the initial conditions, one ob-

serves either a temporal oscillating behavior of  (t ) or a purely stationary partially synchronized regime with  (t ) = const. ∈ (0,1) .

• For K 2 (β, D) ≤ K , only stationary synchronized regimes exist and the fully synchronized case R = 1 is asymptotically reached for K → ∞ .

1.4.3.2  Extensions to more complex noise sources Several generalizations of the basic KS model Eq.(1.98) have recently been discussed, for example with frequency distributions given by: (a) g (ω ) = α ( z )δ (ω − ω 0 ) + (1 − α ( z ))δ (ω + ω 0 ) , this case is studied in [1.36], 1  1 (b) g(ω ) = αδ (ω ) + (1 − α )  δ (ω − ω 0 ) + δ (ω + ω 0 )  , this case is studied in [1.38]. 2  2 The results obtained for both cases (a) and (b) can be reinterpreted as resulting from KS dynamics with specific non-Gaussian noise sources, namely: In case a) the corresponding dynamics will be:

d θk =

K N

N

∑ sin(θ j − θk ) dt + j =1

2 DdZˆ k (t ),



(1.103)

where now Zˆ k (t ) is a biased super-diffusive process which can be represented as the solution to the SDE:

dZˆ k (t ) = (ˆk β)dt + dWk (t ),

Z k (0) = z.

(1.104)

Here ˆk is a biased Bernoulli random variable taking, independently of the Wiener process Wk (t ) , the values +1 and −1 with respective probabilities α( z ) and 1 − α ( z ) where 1 α ( z ) = + tanh (βz ) / 2 . (1.105) 2 The global bifurcation diagram corresponding to this case is given in [1.36]. The asymmetry of g(ω ) reduces the parameter range for which the incoherent behavior is stable and precludes the existence of purely stationary synchronized states, (i.e., states having simultaneously constant  and Θ). The synchronized phases branch off from incoherence as traveling waves thereby implying a time increasing phase Θ(t ) and either a

24

Contemporary Topics In Mathmatics and Statistic Applications

constant amplitude (for coupling strength K close to the bifurcation point), or time oscillating amplitudes (for larger K). In the case (b), the dynamics for oscillator k is:

d θk =

K N

N

∑ sin (θ j − θk )dt + α j =1

2 DdWk (t ) + (1 − α ) 2 DdZ k (t ),

(1.106)

where the parameter α ∈[0,1] is a mixing constant to balance between a non-Gaussian contribution due to (unbiased) dZ k (t ) as defined by Eq.(1.16) and a Gaussian white noise D part dWk (t ). Again, for small deviations from Gaussian noise, characterized by β < , 2 a transition from de-synchronized to a synchronized regime, similar to the pure Gaussian case, arises. For strongly non-Gaussian regimes β > [1.38]:

• • • •

D the following picture emerges 2

4 D < K1 ≤ K ≤ K 2 . Bifurcation to a stationary partially synchronized regime. K 2 ≤ K ≤ K 3 . Strictly positive time-oscillations of (t ) > 0. K3 ≤ K ≤ K 4 . Bi-stable region with possibility, depending on initial conditions, of either time-oscillations of (t ) or pure stationary states with partial synchronization. K 4 < K . Stationary and partially synchronized states with  = const.

The above results clearly exhibit the possibility to generate synchronized stable timeoscillating patterns by a non-Gaussian noise injection and this even in presence of symmetric noise (i.e., vanishing odd moments to any order).

1.4.3.3  Noise induced “zig-zagging’’ - a case study The complex dynamic pattern observed in the previous subsection opens new potentialities for applications. As an illustration, let us study how super-diffusive noise environments may potentially produce spectacular effects in the collective motion of large assemblies of agents such as bacteria, flies, quadrupeds, fishes, etc... . In this context, recently published self-driven particle models have been shown to capture the collective mechanisms for the formation of compact swarms evolving as quasi-solid bodies [1.45], [1.46]-[1.49]. In [1.45], the authors are able to quantify the agents interactions strength leading to the swarms formations. In particular, observations of the barycenter (BC) of a swarm’s collective often reveals “zig-zag” type motions with alternations between traveling in definite direction, almost stopping and restart in a new direction. To unveil one possible mechanism behind this collective dynamics, one now considers simple situations involving N agents, with fixed unit amplitudes velocities, traveling on a plane. In these models, the agents individual direction angles θ k , k = 1, 2,...N will be autonomously updated according to interaction rules dependent on the behavior of ob-

25

On Stochastic Processes Driven by Ballistic Noise Sources

served neighbors. Due to the ubiquitous presence of random fluctuations, noise sources are injected into the dynamics and this obviously produces a tendency to weaken the mechanisms generating collective patterns. This type of modeling is exposed in [1.47] and [1.48]. Therein particles with coupled oscillators dynamics describe the collective behavior of a planar assembly of N agents with positions rk (t ) ∈ (the plane is identified with the complex  -plane) and the orientations θ k (t ) ∈[0, 2π) . The speed being fixed to unity, the assembly dynamics follows:   iθ (t ) (1.107) θ k (t ) = uk r (t ), θ(t ), ξ k (t ) , k = 1,..., N , rk (t ) = e k ,

(

)

where ξ k (t ) are independent stochastic processes and the agents interact via u-control functions: u = (u1 ,..., u N ) . Restricting, similar to [1.50] and [1.51], the u-controls to additive WGN corrupted, pure angle depend functions of the form:

 1  uk θ(t ), dWk (t ) = angle   N

(

)

N

∑e



N

1

N

∑rk (t ) = N ∑e k =1

 + 2 DdWk (t ), 

j =1

with D ∈  + a constant, the swarm’s BC, G (t ) = 1 G (t ) = N

iθ j (t )  

k =1

1 N

N

∑ k =1rk (t ), moves with velocity:

iθk (t )

=: R(t )eiΘ (t ) .

Fig. 1.1: Time evolution of the OP.

(1.108)

26

Contemporary Topics In Mathmatics and Statistic Applications

Fig. 1.2: Time evolution of the swarm’s BC. The BC starts at the origin. At instances of almost full incoherence (OP small) a re-actualization of the global phase changes the velocity direction of the swarm’s BC and produces a zig-zag motion.

Therefore, the speed  (t ) ∈[0,1], in Eq.(1.108) coincides with the OP introduced in Eq.(100). For  = 0, which characterizes the fully incoherent motion (named the balancedphase motion in [1.47]), the swarm’s BC stays fixed in time. When  > 0 the swarm’s BC performs a net motion in direction Θ(t ) . Moreover, from the previous sections we can conclude that for symmetric g(ω ) distributions and appropriate β and K parameters, noise induced temporal modulation of the speed of the swarm’s BC occur. An increase of β reduces the oscillation frequency and increases its amplitude, [1.40]. Remember that for symmetric g(ω ) distributions, oscillations may force the OP to periodically vanish. In such instances of full incoherence a re-actualization of the global phase follows. This changes the velocity direction of the swarm’s BC and ultimately produces a zig-zag motion. This is shown in the two graphs Figure 1.1 and Figure 1.2 where Eq. (1.101) is numerically integrated (Euler’s method, step size h = 0.005 ) with ω k = 0 2 (i.e., g(ω ) = 0 ), N = 7500 , K = 6 , D = 1 and β = ). Figure 1.1 shows the oscillating 2 modulus of the OP. Due to finite size effects, the balanced-phase motion is not completely reached and accordingly the various local minima of R do not completely vanish. The time evolution of the swarm’s BC is captured in Figure 1.2. The evolution of both, the OP and the swarm’s BC are displayed in a sequence of black, grey and red curves, starting at the origin (0, 0) and finishing at the black spot. Color changes synchronously for the BC and the OP every time the OP passes through a local minimum.

On Stochastic Processes Driven by Ballistic Noise Sources

27

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